In the 2nd graph, is there an asymptote?
The definition of a vertical asymptote is important if you want to understand if something is an asymptote or not. From Calculus by Varberg, Purcell, and Rigdon:
That's it. In both graphs you show, we have that statement 3, where $c = 2$, from the definition is true. Since one of those statements is true, the definition says that $x = 2$ is a vertical asymptote in both cases. Notice, the definition doesn't say anything about the value of the function at $x = c$, only the behavior of the graph as $x$ approaches $c$.